page_title: Soddy's hexlet
section_title: Solution by inversion
hierarchical_section_title: Soddy's hexlet / Solution by inversion
context_page_description: In geometry, Soddy's hexlet is a chain of six spheres, each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the blue outer sphere and two spheres above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere, which is not tangent to the three others.
  According to a theorem published by Frederick Soddy in 1937, it is always possible to find a hexlet for any choice of mutually tangent spheres A, B and C. Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres; in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently in Japan, as shown by Sangaku tablets from 1822 in Kanagawa prefecture.
context_section_description: The general problem of finding a hexlet for three given mutually tangent spheres A, B and C can be reduced to the annular case using inversion.  This geometrical operation always transforms spheres into spheres or into planes, which may be regarded as spheres of infinite radius.  A sphere is transformed into a plane if and only if the sphere passes through the center of inversion.  An advantage of inversion is that it preserves tangency; if two spheres are tangent before the transformation, they remain so after.  Thus, if the inversion transformation is chosen judiciously, the problem can be reduced to a simpler case, such as the annular Soddy's hexlet.  Inversion is reversible; repeating an inversion in the same point returns the transformed objects to their original size and position.
  Inversion in the point of tangency between spheres A and B transforms them into parallel planes, which may be denoted as a and b.  Since sphere C is tangent to both A and B and does not pass through the center of inversion, C is transformed into another sphere c that is tangent to both planes; hence, c is sandwiched between the two planes a and b.  This is the annular Soddy's hexlet (Figure 2).  Six spheres s₁–s₆ may be packed around c and likewise sandwiched between the bounding planes a and b.  Re-inversion restores the three original spheres, and transforms s₁–s₆ into a hexlet for the original problem.  In general, these hexlet spheres S₁–S₆ have different radii.
  An infinite variety of hexlets may be generated by rotating the six balls s₁–s₆ in their plane by an arbitrary angle before re-inverting them.  The envelope produced by such rotations is the torus that surrounds the sphere c and is sandwiched between the two planes a and b; thus, the torus has an inner radius r and outer radius 3r.  After the re-inversion, this torus becomes a Dupin cyclide (Figure 3).
caption: Figure 3: A Dupin cyclide, through which the hexlet spheres rotate, always touching. The cyclide is tangent to an inner sphere, an outer sphere and two spheres above and below the "hole" in the "doughnut".
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